54 research outputs found
Two-dimensional droplet spreading over random topographical substrates
We examine theoretically the effects of random topographical substrates on
the motion of two-dimensional droplets via appropriate statistical approaches.
Different random substrate families are represented as stationary random
functions. The variance of the droplet shift at both early times and in the
long-time limit is deduced and the droplet footprint is found to be a normal
random variable at all times. It is shown that substrate roughness decreases
droplet wetting, illustrating also the tendency of the droplet to slide without
spreading as equilibrium is approached. Our theoretical predictions are
verified by numerical experiments.Comment: 12 pages, 5 figure
On the moving contact line singularity: Asymptotics of a diffuse-interface model
The behaviour of a solid-liquid-gas system near the three-phase contact line
is considered using a diffuse-interface model with no-slip at the solid and
where the fluid phase is specified by a continuous density field. Relaxation of
the classical approach of a sharp liquid-gas interface and careful examination
of the asymptotic behaviour as the contact line is approached is shown to
resolve the stress and pressure singularities associated with the moving
contact line problem. Various features of the model are scrutinised, alongside
extensions to incorporate slip, finite-time relaxation of the chemical
potential, or a precursor film at the wall.Comment: 14 pages, 3 figure
A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading
The motion of a contact line is examined, and comparisons drawn, for a
variety of models proposed in the literature. Pressure and stress behaviours at
the contact line are examined in the prototype system of quasistatic spreading
of a thin two-dimensional droplet on a planar substrate. The models analysed
include three disjoining pressure models based on van der Waals interactions, a
model introduced for polar fluids, and a liquid-gas diffuse-interface model;
Navier-slip and two non-linear slip models are investigated, with three
microscopic contact angle boundary conditions imposed (two of these contact
angle conditions having a contact line velocity dependence); and the interface
formation model is also considered. In certain parameter regimes it is shown
that all of the models predict the same quasistatic droplet spreading
behaviour.Comment: 29 pages, 3 figures, J. Eng. Math. 201
The contact line behaviour of solid-liquid-gas diffuse-interface models
A solid-liquid-gas moving contact line is considered through a
diffuse-interface model with the classical boundary condition of no-slip at the
solid surface. Examination of the asymptotic behaviour as the contact line is
approached shows that the relaxation of the classical model of a sharp
liquid-gas interface, whilst retaining the no-slip condition, resolves the
stress and pressure singularities associated with the moving contact line
problem while the fluid velocity is well defined (not multi-valued). The moving
contact line behaviour is analysed for a general problem relevant for any
density dependent dynamic viscosity and volume viscosity, and for general
microscopic contact angle and double well free-energy forms. Away from the
contact line, analysis of the diffuse-interface model shows that the
Navier--Stokes equations and classical interfacial boundary conditions are
obtained at leading order in the sharp-interface limit, justifying the creeping
flow problem imposed in an intermediate region in the seminal work of Seppecher
[Int. J. Eng. Sci. 34, 977--992 (1996)]. Corrections to Seppecher's work are
given, as an incorrect solution form was originally used.Comment: 33 pages, 3 figure
Viscous fluid sheets
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (leaves 108-117).We present a general theory for the dynamics of thin viscous sheets. Employing concepts from differential geometry and tensor calculus we derive the governing equations in terms of a coordinate system that moves with the film. Special attention is given to incorporating inertia and the curvature forces that arise from the thickness variations along the film. Exploiting the slenderness of the film, we assume that the transverse fluid velocity is small compared to the longitudinal one and perform a perturbation expansion to obtain the leading order equations when the center-surface that defines the coordinate system is parametrized by lines of curvature. We then focus on the dynamics of flat film rupture, in an attempt to gain some insights into the sheet breakup and its fragmentation into droplets. By combining analytical and numerical methods, we extend the prior work on the subject and compare our numerical simulations with experimental work reported in the literature.by Nikos Savva.Ph.D
Geometry-induced phase transition in fluids: capillary prewetting
We report a new first-order phase transition preceding capillary condensation
and corresponding to the discontinuous formation of a curved liquid meniscus.
Using a mean-field microscopic approach based on the density functional theory
we compute the complete phase diagram of a prototypical two-dimensional system
exhibiting capillary condensation, namely that of a fluid with long-ranged
dispersion intermolecular forces which is spatially confined by a substrate
forming a semi-infinite rectangular pore exerting long-ranged dispersion forces
on the fluid. In the T-mu plane the phase line of the new transition is
tangential to the capillary condensation line at the capillary wetting
temperature, Tcw. The surface phase behavior of the system maps to planar
wetting with the phase line of the new transition, termed capillary prewetting,
mapping to the planar prewetting line. If capillary condensation is approached
isothermally with T>Tcw, the meniscus forms at the capping wall and unbinds
continuously, making capillary condensation a second-order phenomenon. We
compute the corresponding critical exponent for the divergence of adsorption.Comment: 5 pages, 4 figures, 5 movie
Asymptotic analysis of the evaporation dynamics of partially wetting droplets
We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes
Droplet motion on inclined heterogeneous substrates
We consider the static and dynamic behaviour of two-dimensional droplets on inclined heterogeneous substrates. We utilize an evolution equation for the droplet thickness based on the long-wave approximation of the Stokes equations in the presence of slip. Through a singular perturbation procedure, evolution equations for the location of the two moving fronts are obtained under the assumption of quasi-static dynamics. The deduced equations, which are verified by direct comparisons with numerical solutions to the governing equation, are scrutinized in a variety of dynamic and equilibrium settings. For example, we demonstrate the possibility for stick–slip dynamics, substrate-induced hysteresis, the uphill motion of the droplet for sufficiently strong chemical gradients and the existence of a critical inclination angle beyond which the droplet can no longer be supported at equilibrium. Where possible, analytical expressions are obtained for various quantities of interest, which are also verified by appropriate numerical experiments
Slip or not slip? A methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system
We consider the spreading of a thin two-dimensional droplet on a planar
substrate as a prototype system to compare the contemporary model for contact
line motion based on interface formation of Shikhmurzaev [Int. J. Multiphas.
Flow 19, 589 (1993)], to the more commonly used continuum fluid dynamical
equations augmented with the Navier-slip condition. Considering quasistatic
droplet evolution and using the method of matched asymptotics, we find that the
evolution of the droplet radius using the interface formation model reduces to
an equivalent expression for a slip model, where the prescribed microscopic
dynamic contact angle has a velocity dependent correction to its static value.
This result is found for both the original interface formation model
formulation and for a more recent version, where mass transfer from bulk to
surface layers is accounted for through the boundary conditions. Various
features of the model, such as the pressure behaviour and rolling motion at the
contact line, and their relevance, are also considered in the prototype system
we adopt.Comment: 45 pages, 18 figure
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